How do I properly select part of an object to separate?
The easiest way to select all the vertices in part of a dense mesh, is to switch to wireframe shading (simply press Z or from the 3D view header Viewport Shading > Wireframe) then use a selection tool like border select, to select the region of interest.

The same can be accomplished in solid view by turning off Limit Selection to Visible.


With either method  (wireframe or Limit Selection to Visible) you have disabled the depth culling on the selection, so you can select all the geometry, even the pieces "hiding" behind others.

With the piece selected the way you want then it is a simple matter to separate it P > Selection.